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Theorem ltrncnvatb 30327
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
Hypotheses
Ref Expression
ltrnatb.b  |-  B  =  ( Base `  K
)
ltrnatb.a  |-  A  =  ( Atoms `  K )
ltrnatb.h  |-  H  =  ( LHyp `  K
)
ltrnatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrncnvatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )

Proof of Theorem ltrncnvatb
StepHypRef Expression
1 ltrnatb.b . . . . . 6  |-  B  =  ( Base `  K
)
2 ltrnatb.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 ltrnatb.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
41, 2, 3ltrn1o 30313 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F : B
-1-1-onto-> B )
543adant3 975 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  F : B
-1-1-onto-> B )
6 simp3 957 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  P  e.  B )
7 f1ocnvdm 5796 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( `' F `  P )  e.  B
)
85, 6, 7syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( `' F `  P )  e.  B )
9 ltrnatb.a . . . 4  |-  A  =  ( Atoms `  K )
101, 9, 2, 3ltrnatb 30326 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( `' F `  P )  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
118, 10syld3an3 1227 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( `' F `  P )  e.  A  <->  ( F `  ( `' F `  P ) )  e.  A ) )
12 f1ocnvfv2 5793 . . . 4  |-  ( ( F : B -1-1-onto-> B  /\  P  e.  B )  ->  ( F `  ( `' F `  P ) )  =  P )
135, 6, 12syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( F `  ( `' F `  P ) )  =  P )
1413eleq1d 2349 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( ( F `  ( `' F `  P )
)  e.  A  <->  P  e.  A ) )
1511, 14bitr2d 245 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  B
)  ->  ( P  e.  A  <->  ( `' F `  P )  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290
This theorem is referenced by:  ltrncnvat  30330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-undef 6298  df-riota 6304  df-map 6774  df-plt 14092  df-glb 14109  df-p0 14145  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294
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